First and second order semi-strong interface interaction in multiscale reaction-diffusion systems

Details First and second order semi-strong interface interaction in multiscale reaction-diffusion systems First and second order semi-strong interface interaction in multiscale reaction-diffusion systems

2010-06-22

arxiv.org/abs/1006.4234v3

Nonlinear Sciences

Pattern Formation and Solitons

Scientific paper

In spatial multiscale reaction-diffusion system, where some diffusion lengths are much smaller than the rest, interfaces can form by localisation of only the components with the small diffusion length. The interaction between such interfaces is called semi-strong as it is driven by the non-localised components. The relative velocities of such interfaces are of algebraic order with respect to the diffusion length separation. The quadratic (`second order') case has been intensively studied over the past decade for special models, such as Gierer-Meinhardt and Gray-Scott. The linear (`first order') case was found in the mid-nineties in the Gray-Scott model, but almost no theory has been developped. In this paper a model independent point of view is taken by considering abstract reaction diffusion systems. Starting from a dichotomy in diffusion lengths and assuming existence of interface patterns, characteristic `standard forms' of right hand sides for these two types of interaction are derived formally. The known reaction-diffusion models with semi-strong interface interaction have the corresponding standard form, which is shown explicitly for various examples. The contrasting equations of motion of pulse and front interfaces with first and second order semi-strong interaction are derived. For first order pulse interaction with a single large diffusion length, and under certain natural assumptions, several explicit Lyapunov-functionals such as the largest interpulse distance are found.

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